Abstract
In this paper, we study a stochastic recursive optimal control problem in which the value functional is defined by the solution of a backward stochastic differential equation (BSDE) under G-expectation. Under standard assumptions, we establish the comparison theorem for this kind of BSDE and give a novel and simple method to obtain the dynamic programming principle. Finally, we prove that the value function is the unique viscosity solution to a type of fully nonlinear HJB equation.
Highlights
Soner et al [28] obtained an existence and uniqueness theorem for a new type of backward stochastic differential equation (BSDE) (2BSDE) under a family of nondominated probability measures
As pointed out in [11], the value function defined in (1.3) is a inf sup problem, which is known as the robust optimal control problem
The nonlinear part with respect to ∂x2xV in the HJB equation related to the optimal control problem (1.1)
Summary
Motivated by the model uncertainty in finance, Peng [21–23] established the theory of G-expectation which is a consistent sublinear expectation and does not require a probability space. The representation of G-expectation as the supremum of expectations over a set of nondominated probability measures was obtained in [4, 15]. Due to this set of nondominated probability measures, the backward stochastic differential equation (BSDE for short) is completely different from the classical one. Soner et al [28] obtained an existence and uniqueness theorem for a new type of BSDE (2BSDE) under a family of nondominated probability measures. Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, Shandong 250100, PR China
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